# Distance of a point from a fixed line

If the distance of a point from a fixed line is fixed , why do we say that the locus of the point is a line parallel to the given line ? The point also lies on a line which is perpendicular to the given line right ? Please ask me to elaborate with an example if the statement is not clear. Thank you.

• The word locus here has the significance of all points that satisfy the condition. The problem statement may have been slightly jumbled, but the "given" condition probably involved a line $L$ and a distance $d\gt 0$ from that line. Then one asks about where to find all points $p$ which are that distance from that line. Jan 9 '18 at 13:59
• Actually the locus is a pair of lines, on either sides.
– user65203
Jan 9 '18 at 14:01
• Are you clear what "locus" means? Yes, a point a given distance from a given line lies on many different lines, including the perpendicular to the given line. But the "locus" is the set of all points that distance from the given line. In two dimensions, that locus is a pair of lines. parallel to the given line on either side of it. In three dimensions it is a cylinder. Jan 9 '18 at 14:02
• @user247327 I’m sorry ! I get it now ! All points on the parallel lines would be the same distance from the given line ! Thank you ! Jan 9 '18 at 14:10
• @hardmath thank you ! I understood the concept ! Jan 9 '18 at 14:10

Ok, so excuse the bad drawing. But imagine we tie a rope to a broomstick of length $d$ (for distance). Then the end of this rope (assuming it remain taut) has a few degrees of freedom. It can move along the length of the broomstick (the black line), it can also move around the broomstick. This process, if you were to consider the collection of all places where the green point can go, creates a cylinder around the broom.
If we were to cut right through the middle of this broom we lose one degree of freedom: going "around" the broom. So we're left with two lines whose distance - at every point - from the broom is $d$. This is the definition of a parallel line.
The key understanding here, is that the "locus" is the collection of all points for which a certain condition holds (a distance $d$ from the line)